# On Nonlinear Bending Study of a Piezo-Flexomagnetic Nanobeam Based on an Analytical-Numerical Solution

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

_{1}and u

_{3}, respectively, for axial and transverse directions. However, such displacements for neutral plane are, respectively, regarded with u and w. Thus, one can give accordingly

## 3. Solution Approach

## 4. Numerical Results and Discussion

#### 4.1. Results’ Validity

_{0}a/L = 0. From the Table, it is found that the nondimensional maximum deflection increased as the value of the nonlocal parameter increased.

#### 4.2. Discussion of the Problem

_{31}= 10

^{−9}N/Ampere, f

_{31}= 10

^{−10}N/Ampere as [13,14]. These two values were also theoretically obtained based on some simple assumptions and cannot be the exact numeric values of the flexomagnetic parameter of the aforesaid material presented in Table 3.

_{0}a < 0.8 nm [46], and 0 < e

_{0}a ≤ 2 nm [47,48], unless otherwise stated. The amount of strain gradient parameter was obtained in a similar size to the lattice parameter of the crystalline structure [49]. This factor for the aforementioned material in Table 3 was obtained in an experiment to change between 0.8 and 0.9 nanometers at a set temperature [50]. Hence, the averaged value of the strain gradient parameter is selected as l = 1 nm.

#### 4.2.1. Effect of Nonlinearity

#### 4.2.2. Effect of Small Scale

#### 4.2.3. Effect of Magnetic Field

#### 4.2.4. Effect of Slenderness Ratio

#### 4.2.5. Effect of FM

_{31}= 10

^{−10}N/Ampere can affect to some extent the behavior of the PF-NB.

## 5. Conclusions

- In hinged–hinged nanobeams, linear deflections for a NB can be used in the range w ≤ 0.1 h, and for a PF-NB, about w ≤ 0.08 h. This value in a double-fixed NB and PF-NB is in the range w < 0.15 h. However, for a cantilever case in NB, it is w ≤ 0.2 h and in PF-NB, it is w ≤ 0.1 h.
- The difference between the nonlinear analysis and the linear one will be more pronounced in the boundary condition with higher degrees of freedom.
- Increasing the numerical value of the nonlocal parameter leads to a softening effect on the material, and in contrast, increasing the numerical value of the strain gradient parameter leads to the appearance of stiffness in the material.
- The effect of nonlinear analysis is greater in large values of nonlocal parameters and small values of strain gradient parameters.
- The effect of nonlinear analysis on a nonlocal study is greater than a local one.
- The effect of nonlinear analysis in the positive magnetic field decreases. However, the opposite is true in the case of a negative magnetic field.
- For nanobeams with very large lengths, linear analysis gives entirely erroneous results even if the values of lateral loads are not large.
- The flexomagnetic effect leads to more material stiffness, and thus reduces the numerical values of deflections in static analysis.
- The less flexible the boundary condition, the higher the flexomagneticity effect.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A square (b = h) PF-NB clamped at both ends and exposed to a lateral uniform static loading beside an external magnetic potential.

**Figure 2.**Transverse load vs. different cases of nanobeams (Ψ = 1 mA, l = 1 nm, e

_{0}a = 0.5 nm, C-C).

**Figure 3.**Transverse load vs. different cases of nanobeams (Ψ = 1 mA, l = 1 nm, e

_{0}a = 0.5 nm, S-S).

**Figure 4.**Transverse load vs. different cases of nanobeams (Ψ = 1 mA, l = 1 nm, e

_{0}a = 0.5 nm, C-F).

**Figure 5.**Nonlocal parameter vs. different cases of nanobeams (Ψ = 1 mA, l = 1 nm, p

_{0}= 0.4 N/m, S-S).

**Figure 6.**Nonlocal parameter vs. different cases of nanobeams (Ψ = 1 mA, l = 1 nm, p

_{0}= 0.02 N/m, C-F).

**Figure 7.**Strain gradient parameter vs. different cases of nanobeams (Ψ = 1 mA, e

_{0}a = 1 nm, p

_{0}= 0.4 N/m, C-C).

**Figure 8.**Strain gradient parameter vs. different cases of nanobeams (Ψ = 1 mA, e

_{0}a = 1 nm, p

_{0}= 0.1 N/m, S-S).

**Figure 9.**Magnetic potential parameter vs. different cases of nanobeams (l = 1 nm, e

_{0}a = 0.5 nm, p

_{0}= 0.4 N/m, C-C).

**Figure 10.**Magnetic potential parameter vs. different cases of nanobeams (l = 1 nm, e

_{0}a = 0.5 nm, p

_{0}= 0.1 N/m, S-S).

**Figure 11.**Magnetic potential parameter vs. PF nanobeams (l = 1 nm, e

_{0}a = 0.5 nm, p

_{0}= 0.1 N/m, S-S).

**Figure 12.**Slenderness ratio vs. different cases of nanobeams (Ψ = 1 mA, l = 1 nm, e

_{0}a = 0.5 nm, p

_{0}= 0.4 N/m, C-C).

**Figure 13.**Slenderness ratio vs. different cases of nanobeams (Ψ = 1 mA, l = 1 nm, e

_{0}a = 0.5 nm, p

_{0}= 0.1 N/m, S-S).

**Figure 14.**Transverse load vs. deflection for different cases of nanobeams (Ψ = 1 mA, l = 1 nm, e

_{0}a = 0.5 nm, C-C).

**Figure 15.**Transverse load vs. deflection for different cases of nanobeams (Ψ = 1 mA, l = 1 nm, e

_{0}a = 0.5 nm, S-S).

**Figure 16.**Transverse load vs. deflection for different cases of nanobeams (Ψ = 1 mA, l = 1 nm, e

_{0}a = 0.5 nm, C-F).

**Figure 17.**Presence and absence of flexomagnetic modulus for linear bending of a PF-NB (Ψ = 1 mA, l = 1 nm, e

_{0}a = 0.5 nm, p

_{0}= 0.5 N/m, S-S).

**Table 1.**Dimensionless maximum deflection for a simply-supported nanobeam exposed to transverse uniform loading.

L/h | e_{0}a/L | EBT, Linear [21] | EBT, Linear [45] | EBT, Linear [Present] |
---|---|---|---|---|

10 | 0 | 0.013021 | 0.013021 | 0.013021 |

0.05 | 0.013333 | 0.013333 | 0.013333 | |

0.1 | 0.014271 | 0.014271 | 0.014271 | |

0.15 | 0.015833 | 0.015833 | 0.015833 |

**Table 2.**Maximum deflection (mm) for a clamped–clamped macro beam exposed to transverse uniform loading ($E=210\text{}\mathrm{GPa},\text{}h=5\text{}\mathrm{mm}$).

L/h | p (kN/mm) | EBT, Linear [Present] | FEM, Linear [ABAQUS] |
---|---|---|---|

10 | 0.01 | 0.0792 | 0.0824 |

0.02 | 0.1585 | 0.1648 | |

0.03 | 0.2377 | 0.2472 | |

0.04 | 0.3170 | 0.3297 |

CoFe_{2}O_{4} |
---|

C_{11} = 286 GPa |

q_{31} = 580.3 N/Ampere.m |

a_{33} = 1.57 × 10^{−4} N/Ampere^{2} |

L = 10 h |

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**MDPI and ACS Style**

Malikan, M.; Eremeyev, V.A. On Nonlinear Bending Study of a Piezo-Flexomagnetic Nanobeam Based on an Analytical-Numerical Solution. *Nanomaterials* **2020**, *10*, 1762.
https://doi.org/10.3390/nano10091762

**AMA Style**

Malikan M, Eremeyev VA. On Nonlinear Bending Study of a Piezo-Flexomagnetic Nanobeam Based on an Analytical-Numerical Solution. *Nanomaterials*. 2020; 10(9):1762.
https://doi.org/10.3390/nano10091762

**Chicago/Turabian Style**

Malikan, Mohammad, and Victor A. Eremeyev. 2020. "On Nonlinear Bending Study of a Piezo-Flexomagnetic Nanobeam Based on an Analytical-Numerical Solution" *Nanomaterials* 10, no. 9: 1762.
https://doi.org/10.3390/nano10091762